Particle Traveling in a Uniform Magnetic Field (SR?)

Clearly, it will enter a circular path of radius so and so with angular frequency so and so. But, it will not move in a straight line, not at all. Its momentum will constantly change.

Now, moving to a reference frame moving along the positive [tex]\hat i[/tex] direction with a constant velocity [tex]u[/tex] we find that the force on the particle, given by the Lorentz Force [tex]\vec F_b = q \vec v \times \vec B[/tex] has changed! That means that the change in (Classical) momentum, is not the same in both systems, which is what we would expect!

Focusing on the special case where [tex]u=v_0[/tex] we find that in the primed reference frame, there is no change in momentum at all!

I've done some reading and found that this apparent contradiction is reconciled by considering that the EM field is transformed relativistically and produces a new EM field that provides adequate results.

I could, however, only find quantitative analysis of the phenomenon (The actual transforms themselves) and qualitative analysis for a very specific situation (Current carrying wire) where the source of the new EM field was explained by length contraction and the formation of a net charge density. How this applies to the charge free region of space my question refers to eludes me.

A link or referral to a source describing the rationale behind the transforms would be much appreciated. (I only have the most basic understanding of SR and a moderate understanding of EM, so something appropriate would be wonderful)

Another thing that's been bothering me, is that this point to reflect on was given in our first lesson on Magnetism at school (The introduction to the Lorentz Force). Though it was unclear if we're supposed to be able and answer it (No one in my class has any knowledge of SR).